Pointed encompassing trees

It is shown that for any set of disjoint line segments in the plane there exists a pointed binary encompassing tree, that is, a spanning tree on the segment endpoints that contains all input segments, has maximal degree three, and such that every vertex is incident to an angle greater than π. As a consequence, it follows that every set of disjoint line segments has a bounded degree pseudo-triangulation

Detecting Weakly Simple Polygons

A closed curve in the plane is weakly simple if it is the limit (in the Fr\'echet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We also describe algorithms that detect weak simplicity in O(n log n) time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.Comment: 25 pages and 13 figures, submitted to SODA 201

Modal logics are coalgebraic

Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of specific logics used in particular domains. The coalgebraic approach is generic and compositional: tools and techniques simultaneously apply to a large class of application areas and can moreover be combined in a modular way. In particular, this facilitates a pick-and-choose approach to domain specific formalisms, applicable across the entire scope of application areas, leading to generic software tools that are easier to design, to implement, and to maintain. This paper substantiates the authors' firm belief that the systematic exploitation of the coalgebraic nature of modal logic will not only have impact on the field of modal logic itself but also lead to significant progress in a number of areas within computer science, such as knowledge representation and concurrency/mobility

Hurwitz equivalence of braid monodromies and extremal elliptic surfaces

We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $\Gamma$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\it topologically} distinct algebraic objects such as extremal elliptic surfaces, real trigonal curves, and real elliptic surfaces

The Assembly and Merging History of Supermassive Black Holes in Hierarchical Models of Galaxy Formation

We assess models for the assembly of supermassive black holes (SMBHs) at the center of galaxies that trace their hierarchical build-up far up in the dark halo `merger tree'. We assume that the first `seed' black holes (BHs) formed in (mini)halos collapsing at z=20 from high-sigma density fluctuations. As these pregalactic holes become incorporated through a series of mergers into larger and larger halos, they sink to the center owing to dynamical friction, accrete a fraction of the gas in the merger remnant to become supermassive, form a binary system, and eventually coalesce. The merger history of dark matter halos and associated BHs is followed by cosmological Monte Carlo realizations of the merger hierarchy. A simple model, where quasar activity is driven by major mergers and SMBHs accrete at the Eddington rate a mass that scales with the fifth power of the velocity dispersion, is shown to reproduce the optical LF of quasars in the redshift range 1<z<4. Binary and triple BH interactions are followed in our merger tree. The assumptions underlying our scenario lead to the prediction of a population of massive BHs wandering in galaxy halos and the intergalactic medium at the present epoch, and contributing <10% to the total BH mass density. At all epochs the fraction of binary SMBHs in galaxy nuclei is of order 10%, while the fraction of binary quasars is less than 0.3%Comment: revised version, accepted for publication in the ApJ, emulateapj, 15 pages, 16 figure